1. bookVolume 79 (2021): Edition 2 (December 2021)
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12 Nov 2012
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Certain Singular Distributions and Fractals

Publié en ligne: 01 Jan 2022
Volume & Edition: Volume 79 (2021) - Edition 2 (December 2021)
Pages: 163 - 198
Reçu: 01 Jan 2021
Détails du magazine
License
Format
Magazine
eISSN
1338-9750
Première parution
12 Nov 2012
Périodicité
3 fois par an
Langues
Anglais
Abstract

In the presented paper, the main attention is given to fractal sets whose elements have certain restrictions on using digits or combinations of digits in their own nega-P-representation. Topological, metric, and fractal properties of images of certain self-similar fractals under the action of some singular distributions, are investigated.

Keywords

[1] DE AMO, E.—CARRILLO, M.D. —FERNÁNDEZ-SÁNCHEZ, J.: A Salem generalized function, Acta Math. Hungar. 151 (2017), no. 2, 361–378. https://doi.org/10.1007/s10474-017-0690-x10.1007/s10474-017-0690-x Search in Google Scholar

[2] BALANKIN, A.S.—BORY REYES, J.—LUNA-ELIZARRARÁS, M.E. —SHAPIRO, M.: Cantor-type sets in hyperbolic numbers, Fractals 24 (2016), no. 4, Paper no. 1650051. Search in Google Scholar

[3] BRODERICK, R.—FISHMAN, L.—REICH, A.: Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler, Mosc. J. Comb. Number Theory 1 (2011), no. 4, 3–12. Search in Google Scholar

[4] BUNDE, A.—HAVLIN, S.: Fractals in Science, Springer-Verlag, Berlin, 1994.10.1007/978-3-642-77953-4 Search in Google Scholar

[5] DANI, S.G.—SHAH, HEMANGI: Badly approximable numbers and vectors in Cantor-like sets, Proc. Amer. Math. Soc. 140 (2012), 2575–2587.10.1090/S0002-9939-2011-11105-5 Search in Google Scholar

[6] DIMARTINO, R.—URBINA, W. O.: On Cantor-like sets and Cantor-Lebesgue singular functions, https://arxiv.org/pdf/1403.6554.pdf Search in Google Scholar

[7] DIMARTINO, R.—URBINA, W. O.: Excursions on Cantor-like Sets, https://arxiv.org/pdf/1411.7110.pdf Search in Google Scholar

[8] FALCONER, K.: Techniques in Fractal Geometry, John Willey and Sons, Ltd., Chichester, 1997.10.2307/2533585 Search in Google Scholar

[9] FALCONER, K.: Fractal Geometry: Mathematical Foundations and Applications, Wiley, 2004.10.1002/0470013850 Search in Google Scholar

[10] FENG, D. J.: The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers, Adv. Math. 195 (2005), 24–101.10.1016/j.aim.2004.06.011 Search in Google Scholar

[11] HUA, S.—RAO, H.—WEN, Z. ET AL.: textitOn the structures and dimensions of Moran sets, Sci. China Ser. A-Math. 43 (2000), no. 8, 836–852 Search in Google Scholar

[12] ITO, S.—SADAHIRO, T.: Beta-expansions with negative bases, Integers 9 (2009), 239–259.10.1515/INTEG.2009.023 Search in Google Scholar

[13] KÄENMÄKI, A.—LI, B.—SUOMALA, V.: Local dimensions in Moran constructions, Nonlinearity 29 (2016), no. 3, 807–822. Search in Google Scholar

[14] KALPAZIDOU, S.—KNOPFMACHER, A.—KNOPFMACHER, J.: Lüroth-type alternating series representations for real numbers, Acta Arithmetica 55 (1990), 311–322.10.4064/aa-55-4-311-322 Search in Google Scholar

[15] KATSUURA, H.: Continuous nowhere-differentiable functions - an application of contraction mappings, Amer. Math. Monthly 98 (1991), no. 5, 411–416, https://doi.org/10.1080/00029890.1991.1200077810.1080/00029890.1991.12000778 Search in Google Scholar

[16] KAWAMURA, K.: The derivative of Lebesgue’s singular function, In: Summer Symposium 2010, Real Anal. Exchange, pp. 83–85. Search in Google Scholar

[17] KENNEDY, J. A.—YORKE, J. A.: Bizarre topology is natural in dynamical systems, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 3, 309–316. Search in Google Scholar

[18] LI, J.— WU, M.: Pointwise dimensions of general Moran measures with open set condition, Sci. China, Math. 54 (2011), 699–710. Search in Google Scholar

[19] MANDELBROT, B.: Fractals: Form, Chance and Dimension. W. H. Freeman and Co., San Francisco, Calif. 1977. Search in Google Scholar

[20] MANDELBROT, B.: The Fractal Geometry of Nature. 18th printing, Freeman, New York, 1999. Search in Google Scholar

[21] MORAN, PA. P.: Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), no. 1, 15–23, doi:10.1017/S0305004100022684.10.1017/S0305004100022684 Search in Google Scholar

[22] PALIS, J.—TAKENS, F.: Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors. In: Cambridge Studies in Advanced Mathematics Vol. 35, Cambridge University Press, Cambridge, 1993. Search in Google Scholar

[23] POLLICOTT, M.—SIMON, K.: The Hausdorff dimension of λ-expansions with deleted digits, Trans. Amer. Math. Soc. 347 (1995), no. 3, 967–983. https://doi.org/10.1090/S0002-9947-1995-1290729-010.1090/S0002-9947-1995-1290729-0 Search in Google Scholar

[24] RÉNYI, A.: Representations for real numbers and their ergodic properties, Acta. Math. Acad. Sci. Hungar. 8 (1957), 477–493.10.1007/BF02020331 Search in Google Scholar

[25] SALEM, R.: On some singular monotonic functions which are stricly increasing, Trans. Amer. Math. Soc. 53 (1943), 423–439.10.1090/S0002-9947-1943-0007929-6 Search in Google Scholar

[26] SERBENYUK, S. O.: Topological, metric and fractal properties of one set defined by using the s-adic representation. In: XIV International Scientific Kravchuk Conference: Conference materials II, Kyiv: National Technical University of Ukraine “KPI”, 2012, p.220. (In Ukrainian) https://www.researchgate.net/publication/311665455 Search in Google Scholar

[27] SERBENYUK, S.O.: Topological, metric and fractal properties of sets of class generated by one set with using the s-adic representation. In: International Conference “Dynamical Systems and their Applications: Abstracts, Kyiv: Institute of Mathematics of NAS of Ukraine, 2012, p. 42. (In Ukrainian) https://www.researchgate.net/publication/311415778 Search in Google Scholar

[28] SERBENYUK, S. O.: Topological, metric and fractal properties of the set with parameter, that the set defined by s-adic representation of numbers. In: International Conference “Modern Stochastics: Theory and Applications III dedicated to 100th anniversary of B. V. Gnedenko and 80th anniversary of M. I. Yadrenko: Abstracts, Kyiv: Taras Shevchenko National University of Kyiv, 2012, p. 13. https://www.researchgate.net/publication/311415501 Search in Google Scholar

[29] SERBENYUK, S.O.: Topological, metric, and fractal properties of one set of real numbers such that it defined in terms of the s-adic representation, Naukovyi Chasopys NPU im. M. P. Dragomanova. Seria 1. Phizyko-matematychni Nauky [Trans. Natl. Pedagog. Mykhailo Dragomanov University. Ser. 1. Phys. Math.] 11 (2010), 241–250. (in Ukrainian) https://www.researchgate.net/publication/292606441 Search in Google Scholar

[30] SERBENYUK, S. O.: Topological, metric properties and using one generalizad set determined by the s-adic representation with a parameter, Naukovyi Chasopys NPU im. M.P. Dragomanova. Ser. 1. Phizyko-matematychni Nauky [Trans. Natl. Pedagog. Mykhailo Dragomanov University. Ser. 1. Phys. Math.] 12 (2011), 66–75. (In Ukrainian) https://www.researchgate.net/publication/292970196 Search in Google Scholar

[31] SERBENYUK, S. O.: On some sets of real numbers such that defined by nega-s-adic and Cantor nega-s-adic representations, Naukovyi Chasopys NPU im. M. P. Dragomanova. Seria 1. Phizyko-matematychni Nauky [Trans. Natl. Pedagog. Mykhailo Dragomanov Univ. Ser.1. Phys. Math.] 15 (2013), 168–187. (In Ukrainian) https://www.researchgate.net/publication/292970280 Search in Google Scholar

[32] SERBENYUK, S. O.: Functions, that defined by functional equations systems in terms of Cantor series representation of numbers, Naukovi Zapysky NaUKMA 165 (2015), 34–40. (In Ukrainian) https://www.researchgate.net/publication/292606546 Search in Google Scholar

[33] SERBENYUK, S.: On some generalizations of real numbers representations, arXiv:1602.07929v1. (In Ukrainian) Search in Google Scholar

[34] SERBENYUK, S.: One one class of fractal sets, https://arxiv.org/pdf/1703.05262.pdf Search in Google Scholar

[35] SERBENYUK, S.: More on one class of fractals, arXiv:1706.01546v1. Search in Google Scholar

[36] SERBENYUK, S.O.: One distribution function on the Moran sets, Azerb. J. Math. 10 (2020), no. 2, 12–30, arXiv:1808.00395v1. Search in Google Scholar

[37] SERBENYUK, S.: Nega-Q˜\[\tilde Q\]-representation as a generalization of certain alternating representations of real numbers, Bull. Taras Shevchenko Natl. Univ. Kyiv Math. Mech. 1 (35) (2016), no.1, 32–39. (In Ukrainian) https://www.researchgate.net/publication/308273000 Search in Google Scholar

[38] SERBENYUK, S.: On one class of functions with complicated local structure, Šiauliai Mathematical Seminar 11 (19) (2016), 75–88. Search in Google Scholar

[39] SERBENYUK, S. O.: Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers, Zh. Mat. Fiz. Anal. Geom. 13 (2017), no. 1, 57–81. https://doi.org/10.15407/mag13.01.05710.15407/mag13.01.057 Search in Google Scholar

[40] SERBENYUK, S. O.: Non-differentiable functions defined in terms of classical representations of real numbers, Zh. Mat. Fiz. Anal. Geom. 14 (2018), no. 2, 197–213. https://doi.org/10.15407/mag14.02.19710.15407/mag14.02.197 Search in Google Scholar

[41] SERBENYUK, S. O.: Preserving the Hausdorff-Besicovitch dimension by monotonic singular distribution functions. In: Second Interuniversity Scientific Conference on Mathematics and Physics for Young Scientists: Abstracts. Institute of Mathematics of NAS of Ukraine, Kyiv (2011). pp. 106–107. (In Ukrainian) https://www.researchgate.net/publication/301637057 Search in Google Scholar

[42] SERBENYUK, S.: On one fractal property of the Minkowski function, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Ser. A. Matematicás 112 (2018), no. 2, 555–559, doi:10.1007/s13398-017-0396-5.10.1007/s13398-017-0396-5 Search in Google Scholar

[43] SERBENYUK, S.: On one application of infinite systems of functional equations in function theory, Tatra Mt. Math. Publ. 74 (2019), 117–144. https://doi.org/10.2478/tmmp-2019-002410.2478/tmmp-2019-0024 Search in Google Scholar

[44] SERBENYUK, S.: On certain maps defined by infinite sums, J. Anal. 28 (2020), no. 4, 987–1007. https://doi.org/10.1007/s41478-020-00229-x10.1007/s41478-020-00229-x Search in Google Scholar

[45] TAYLOR, T. D.—HUDSON, C.— ANDERSON, A.: Examples of using binary Cantor sets to study the connectivity of Sierpinski relatives, Fractals 20 (2012), no. 1, 61–75. Search in Google Scholar

[46] TÉLLEZ-SÁNCHEZ, G. Y.—BORY-REYES, J.: More about Cantor like sets in hyperbolic numbers, Fractals 25 (2017), no. 5, Paper no. 1750046. Search in Google Scholar

[47] WANG, B. W.— WU, J.: Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math. 218 (2008), 1319–1339.10.1016/j.aim.2008.03.006 Search in Google Scholar

[48] WIKIPEDIA CONTRIBUTORS: Fractal, Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Fractal Search in Google Scholar

[49] WIKIPEDIA CONTRIBUTORS: Pathological (mathematics), Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Pathological_(mathematics) Search in Google Scholar

[50] WIKIPEDIA CONTRIBUTORS: Self-similarity, Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Self-similarity Search in Google Scholar

[51] WIKIPEDIA CONTRIBUTORS: Singular function, Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Singular_function Search in Google Scholar

[52] WIKIPEDIA CONTRIBUTORS: Thomae’s function, Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Thomae’s_function Search in Google Scholar

[53] WU, J.: On the sum of degrees of digits occurring in continued fraction expansions of Laurent series, Math. Proc. Camb. Philos. Soc. 138 (2005), 9–20.10.1017/S0305004104008163 Search in Google Scholar

[54] WU, M.: The singularity spectrum f (α) of some Moran fractals, Monatsh. Math. 144 (2005), 141–155.10.1007/s00605-004-0254-3 Search in Google Scholar

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